Characteristic and Minimal polynomials of matrices

In summary, the conversation discusses a linear map and its matrix representation, and the task of finding the power b such that the characteristic polynomial equals (X-6)^b. The person attempted to use Mathematica to find b, but was unsuccessful. They also mention struggling to find a basis for the generalized eigenspace Vt(6).
  • #1
jiles-smith
7
0

Homework Statement



Let V=C^4 and consider the linear map V->V given by the matrix:

{{12,-6,6,-6},{2,21,21,51},{-3,12,12,30},{1,-9,-9,-21}}

(Each {...} denotes a row, tried to use Latex but got extremely confused!)

Given that chA(X)=(X-6)^4, calculate:

(i) The power such that mA(X)=(X-6)^b
(ii) A basis for the generalized eigenspace Vt(6) where t=1,...,b


The Attempt at a Solution


I attempted to find b by sticking mA(A) into mathematica and seeing if there is a power such that mA(A)=0. This didn't work at all. Any help would be much appreciated.
 
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  • #2
For (ii) I know that the generalised eigenspace is the null space of A-6I but I'm struggling to find a basis for it.
 

Related to Characteristic and Minimal polynomials of matrices

1. What is a characteristic polynomial of a matrix?

A characteristic polynomial of a matrix is a polynomial function that is used to find the eigenvalues of a square matrix. It is defined as det(A - xI), where A is the given matrix and x is a variable. The roots of this polynomial are the eigenvalues of the matrix.

2. How is the characteristic polynomial related to the minimal polynomial?

The minimal polynomial of a matrix is the smallest degree polynomial that when substituted with the matrix, gives a zero matrix. It is a divisor of the characteristic polynomial, meaning that the roots of the minimal polynomial are also roots of the characteristic polynomial. However, the converse is not always true.

3. How do you find the characteristic and minimal polynomials of a matrix?

To find the characteristic polynomial, you first need to find the determinant of (A - xI), where A is the given matrix and x is a variable. The characteristic polynomial will be a polynomial function of x. To find the minimal polynomial, you need to find the largest degree polynomial that when substituted with the matrix, gives a zero matrix. This can be done through various methods such as using the Cayley-Hamilton theorem or using the concept of invariant factors.

4. What is the significance of characteristic and minimal polynomials in matrix computations?

The characteristic polynomial helps in finding the eigenvalues of a matrix, which is a crucial step in solving systems of linear equations, determining matrix similarity, and finding diagonalizable matrices. The minimal polynomial helps in determining the Jordan canonical form of a matrix, which has applications in solving differential equations and studying linear transformations.

5. Can a matrix have more than one minimal polynomial?

Yes, a matrix can have multiple minimal polynomials. This is because the minimal polynomial is not unique and can depend on the field over which the matrix is defined. For example, a matrix may have different minimal polynomials over the real numbers and complex numbers.

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